On the Number of Representations of Integers by various Quadratic and Higher Forms

We give formulas for the number of representations of non negative integers by various quadratic forms. We also give evaluations in the case of sum of two cubes (cubic case) and the quintic case, as well. We introduce a class of generalized triangular numbers and give several evaluations. Finally, we present a mean value asymptotic formula for the number of representations of an integer as sum of two squares known as the Gauss circle problem.